Abstract
In this chapter a formulation of Boolean networks is introduced and their topological structure is then investigated. The matrix expression of logic, discussed in previous chapters, is used and, under this framework, the dynamics of a Boolean network equation is converted into an equivalent algebraic form as a standard discrete-time linear system. Analyzing the transition matrix of the linear system, easily computable formulas are obtained to show (a) the number of fixed points, (b) the numbers of cycles of different lengths, (c) the transient period, which is the time for all points to enter the set of attractors, (d) the basin of each attractor. In addition, algorithms are developed to calculate all the fixed points, cycles, transient periods, and basins of attraction of all attractors. This chapter is partly based on Cheng and Qi (IEEE Trans. Automat. Contr. 55(10):2251–2258, 2010).
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Cheng, D., Qi, H., Li, Z. (2011). Topological Structure of a Boolean Network. In: Analysis and Control of Boolean Networks. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-097-7_5
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DOI: https://doi.org/10.1007/978-0-85729-097-7_5
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